Up the creek. What you are looking for is not that uncommon here in Maine. Wood fired furnaces and aux wood fired systems are getting more popular. Many are run together with another system such as you are looking to do. I am no expert on these things, but have two good friends that have put in external wood furnaces and had them plumbed into the domestic hot water and forced hot water heating system. The key to the system is using a wood stove/furnace with a properly designed heat exchanger that easily works with your existing system. The theory is simple, wood preheats water going into your boiler. The hotter the preheated water is, the less your gas furnace has to work.
I saw several different systems at a home show last spring. If you don't have any HVAC experience, I'd really suggest checking out ready made systems at your local dealers. Creating something like this from scratch looks pretty difficult to me. A good dealer can show you what you will need and what options would work with your situation. Safety is always your first priority, so get some good advice, good equipment and you should be on the way to saving a lot of money in the long run. Good luck...
Particularly if you have an oversized boiler already, extending the hot water makes more sense.
It's not terribly difficult to insulate it properly - especially for a mere 10 or 15 feet. Use 1" PEX and build an XPS (waterproof styrofoam insulation sheets) box, keeping the lines (supply and return) separated with insulation, polyurethane foam it to fill and seal, done (and done better than buying the absurd products sold with two lines in a 4" tube with not nearly enough insulation around or between them, for stupid high prices.)
This link to a more-specifically heating-related forum may provide some insight.
If you extend the gas line instead, you need another device to burn gas, venting for it, etc.
If you extend the hot water, all you need is radiators/baseboard/etc. and you do NOT have an ignition source or a fuel source in your piano-storage-barn, which may positively impact your insurance (if any) and/or peace of mind.
Here is picture of a (much larger than you need, industrial-style and steam-pipe carrying) bridge at the Wright Brother's Museum
Effectively you should have something that looks like a 15 foot long box, probably 12" square, between the buildings - rather than exposed steel as in the picture, wood or siding/roofing to match your buildings. Or it can simply look like a large metal pipe or duct, which is one way to deal with keeping it dry when not trying to match the roofing/siding. As mentioned WRT the burial depth, a long power outage causing loss of heating in the winter will be more prone to freezing this type of arrangement than one that's well-buried. Personally, I'd bury it, it's not that hard or expensive to do, and it's invisible.
Best Answer
In the US, hot water is often generated by the furnace. These specially designed units have a dedicated coil in the firebox that allows cold water in one end, circulates it through the coil and expels hot water from the other. The benefit (as you suggest) is the use of just one device for both room and water heating. The disadvantage is the main furnace is kept active, even in warm weather, to ensure ongoing access to hot water, a slightly less efficient system.
In theory, you could create a copper coil to run water through a heated area of your stove. However, this presents numerous complex problems of temperature, routing, sealing, all of which are beyond the skill of most dedicated DIYers. The risks are gas leaks, fire, explosion, water leaks, flood, and a plague of locusts.
Don't try it!